# Fermi Level Calculator

Created by Dominik Czernia, PhD
Reviewed by Małgorzata Koperska, MD and Steven Wooding
Last updated: Feb 02, 2023
Table of contents:

Our Fermi level calculator is a handy tool that helps you to determine Fermi parameters from the number density of electrons. Moreover, you can use this calculator to compute the probability of finding a particle with specific energy, which is the so-called Fermi function (the Fermi-Dirac statistics).

Read on if you want to learn what a fermi level is and how to calculate it. In the following text, you will also find Fermi level equations that can be used to calculate:

• Fermi energy;
• Fermi temperature;
• Fermi velocity; and
• Fermi wave vector (Fermi wave number).

Finally, we have provided a table of the number density of free electrons in some metals, which you can use in this calculator.

## What is the Fermi level? – Fermi level definition

The Fermi energy is a physical quantity that is used to determine the maximum energy of an electron at absolute zero temperature (0 K). All energy levels up to the Fermi level are occupied by electrons, and all levels above are unoccupied (you can read more about energy levels in Bohr model calculator).

The existence of the Fermi level is a consequence of the Pauli exclusion principle, which states that two fermions, e.g., electrons, cannot be in the same stationary state. Fermions are particles that obey Fermi-Dirac statistics and have spin $\small \rm s = 1/2$. You should check out our magnetic moment calculator to read more about spin and the quantum number calculator to learn about quantum numbers.

In other words, the Fermi level is the surface (sometimes called Fermi surface) at absolute zero temperature, below which no electrons will have enough energy to rise above the surface. To find Fermi energy, we can imagine that we have an empty system, which we want to fill with electrons. We add electrons one at a time, which will consecutively take further places on stationary states with the lowest energy. When all the electrons have been put in, the highest electron energy level is Fermi energy.

However, at higher temperatures, a certain fraction of electrons will have high enough energy to exist above the Fermi level. The number of those electrons can be estimated using the Fermi-Dirac distribution.

## Calculate Fermi energy, Fermi temperature, Fermi velocity and Fermi wave vector (Fermi wave number)

Although the above Fermi definition is theoretically correct, we actually use the Fermi gas model to quantify Fermi parameters. A Fermi gas is a phase of matter which consists of a large number of non-interacting fermions. In this case, the Fermi surface forms the Fermi sphere, above which no electron can rise at the absolute zero temperature. You can use our Fermi level calculator to quickly compute Fermi parameters with the following Fermi level equations:

1. Fermi wave vector (Fermi wave number) $k_{\rm F}$:
$\qquad k_{\rm F} = \sqrt[3]{3\pi^2n}$
1. Fermi energy $E_{\rm F}$:
$\qquad E_{\rm F} = \hbar^2\frac{k_{\rm F}^2}{2m_e}$
1. Fermi velocity $v_{\rm F}$:
$\qquad v_{\rm F} = \hbar \frac{k_{\rm F}}{m_e}$
1. Fermi temperature $T_{\rm F}$:
$\qquad T_{\rm F} = \frac{E_{\rm F}}{k_{\rm B}}$

In these equations:

• $n$ – Electron number density (number of electrons per unit volume);
• $m_e$ – Electron mass $\small m_e = 9.10938356\! \times\! 10^{-31}\ \rm kg$;
• $k_{\rm B}$ – Boltzmann constant $\small k_{\rm B} = 1.38064852\! \times\! 10^{-23}\ \rm m^2\! \cdot\! kg / (s^2\! \cdot\! K)$; and
• $\hbar$ – Reduced Plank constant $\small \hbar = h/2\pi = 1.054571817\! \times\! 10^{-34}\ \rm J\! \cdot\! s$.

Did you know that you can express energy in many units? Not only in joules but also calories and kilo-watt hours or even in kelvins and kilograms! Check our energy conversion to find out how you can compute energy in different units.

## The Fermi-Dirac distribution

The Fermi-Dirac distribution (or Fermi-Dirac statistics) is the probability that an electron is in a specific energy state (energy level). It applies to identical, indistinguishable particles with half-integer spin (fermions), in particular electrons. Additionally, those particles don't interact with each other (Fermi gas model).

Our Fermi level calculator uses the following Fermi function definition:

$f(E) = \frac{1}{e^{(E - E_{\rm F})/k_{\rm B}T}+1},$

where:

• $f(E)$ – Probability that a particle will have energy $E$;
• $e$ – Mathematical constant $\small e \approx 2.71828$,
• $E$ – Energy of a particle; and
• $T$ – Temperature.

As we stated in the previous sections, at the absolute zero temperature, every single energy level below the Fermi surface will be occupied, and every energy level above the Fermi surface will be vacant. You can check whether this is true with our Fermi level calculator. Increasing temperature will change the situation – some electrons will rise above the Fermi energy. Nevertheless, if the energy of the particle equals Fermi energy $\small E = E_{\rm F}$, then $\small f(E)$ should be exactly 0.5 regardless of temperature.

## Free electron number densities table

Most metals are good electrical conductors – we use this fact in electronics. We can describe the behavior of their electrons very well with the so-called free electron model. It considers electrons as an electron gas with no mutual interactions between electrons (analogically to Fermi gas).

In the table below, you can find the number densities of some typical metals (or you can use number density calculator to compute it yourself), which can be used in our Fermi level calculator. Just select any material.

Metal

Number density (×10²⁸ electrons/m³)

Cu (copper)

8.47

Ag (silver)

5.86

Au (gold)

5.90

Be (beryllium)

24.7

Mg (magnesium)

8.61

Ca (calcium)

4.61

Sr (strontium)

3.55

Ba (barium)

3.15

Nb (niobium)

5.56

Fe (iron)

17.0

Zn (zinc)

13.2

Cd (cadmium)

9.27

Al (aluminum)

18.1

Ga (gallium)

15.4

In (indium)

11.5

Sn (tin)

14.8

Pb (lead)

13.2

Dominik Czernia, PhD
Material properties
Material
Select...
Number density of electrons
×10²⁸ electrons/
cu yd
Fermi parameters
Fermi wave number
nm
⁻¹
Fermi energy
eV
Fermi temperature
K
Fermi velocity
ft/s
Fermi-Dirac distribution
Energy
eV
Temperature
K
Fermi Function
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